Big and small numbers in physics - Group task
Work in groups to try to create the best approximations to these
physical quantities.
Age
16 to 18
Challenge level
Problem
Physics makes use of numbers both small and large. Try these questions involving big and small numbers. You might need to use pieces of physical data not given in the question. Sometimes these questions involve estimation, so there will be no definitive 'correct' answer; on other occasions an exact answer will be appropriate. Use your judgement as seems appropriate in each context. Feel free to attempt them in any order; some will seem easier than other dependent on your knowledge of physics.
Your goal is to provide the best, sensible approximation to the questions taking into account the precision to which each question is stated. Along with finding a numerical answer, clearly express any scientific or modelling assumptions made and which formulae you used along the way.
- It is known that the value of $g$ on the moon is about one-sixth that on earth. How high do you think that you would be able to jump straight up on the surface of the moon?
- The mass of an atom of lead is $3.44\times 10^{-22}$g. Lead has a density of $11.35$ g cm$^{-1}$. How many atoms of lead are found in a single cubic centimetre of lead?
- The earth orbits the sun on an almost circular path of average radius about $149\,598\,000\,000$m. How fast is the earth moving relative to the sun?
- The tallest buildings in the world are over $800$m high. If I dropped a cricket ball off the top of one of these, estimate how fast it would be moving when it hit the ground.
- What weight of fuel would fit into a petrol tanker?
- The charge on a proton is $1.6\times 10^{-19}$C. What is the total sum of the positive charges in a litre of Hydrochloric acid of pH 1.0?
- What is the mass of a molecule of water?
- How many molecules of water are there in an ice cube?
- Around 13.4 billion years ago the universe became sufficiently cool that atoms formed and photons present at that time could propagate freely (this time was called the surface of last scattering). How far would one of these old photons have travelled by now?
- How much energy is contained in the matter forming the earth?
NOTES AND BACKGROUND
An obvious part of the skill with applying mathematics to physics is to know the fundamental formulae and constants relevant to a problem. By not providing these pieces of information directly, you need to engage at a deeper level with the problems. You might not necessarily know all of the required formulae, but working out which parts you can and cannot do is all part of the problem solving process!
Approximation problems can involve sophisticated application of mathematics, especially when clearly stated in the form: given these assumptions, the following numerical consequences follow.
Getting Started
Note that most of the ideas used here are typically covered at school before the age of 16, although possibly in mathematics, physics or chemistry.
In estimation questions don't be afraid to have a go with a guess at some numbers in the problem and then to refine your estimate after checking it makes some sort of sense.
Although there is no 'right' answer to an estimation, there are good or bad estimates and sensible or over detailed calculations.
Think how you might make your estimation a good one, and think how it makes sense to ignore certain complexities in particular calculations.
Teachers' Resources
The notes below describe a method of engagement based around a technique called complex instruction.
Although this problem is group-worthy, it can, of course, be attempted individually if you wish.
Why do this problem?
This problem lends itself to collaborative working, both for students who are inexperienced at working in a group and students who are used to working in this way.
The problem involves working out sensible approximations to physical quantities. Making good headway into the task will be difficult and requires good use of teamwork.
Many NRICH tasks have been designed with group work in mind. Here we have gathered together a collection of short articles that outline the merits of collaborative work, together with examples of teachers' classroom practice.
Possible approach
This is an ideal problem for students to tackle in groups of four. Allocating these clear roles (Word, pdf) can help the group to work in a purposeful way - success on this task should be measured by how effectively the members of the group work together as well as by the solutions they reach.
Introduce the four group roles to the class. It may be appropriate, if this is the first time the class have worked in this way, to allocate particular roles to particular students. If the class work in roles over a series of lessons, it is desirable to make sure everyone experiences each role over time.
For suggestions of team-building maths tasks for use with classes unfamiliar with group work, take a look at this article and the accompanying resources.
You may want to make calculators, internet, squared or graph paper, poster paper, and coloured pens available for the Resource Manager in each group to collect. As teacher, you (or the internet) will be a resource containing knowledge of physical data, constants and formulae.
While groups are working divide the board up with the groups names as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together. This is a good way of highlighting the mathematical behaviours you want to promote, particularly with a challenging task such as this.
You may choose to focus on the way the students are co-operating:
Group A - Good to see you sharing different ways of thinking about the problem.
Group B - I like the way you are keeping a record of people's ideas and results.
Group C - Resource manager - is there anything your team needs? Are there any facts or data that you need but don't know?
Group B - I like the way you are keeping a record of people's ideas and results.
Group C - Resource manager - is there anything your team needs? Are there any facts or data that you need but don't know?
Alternatively, your focus for feedback might be mathematical:
Group A - I like the way you set out your assumptions clearly.
Group B - Can you provide some error bounds on that calculation?
Group C - Good to see that someone's checking the numerical calculations.
Group B - Can you provide some error bounds on that calculation?
Group C - Good to see that someone's checking the numerical calculations.
Make sure that while groups are working they are reminded of the need to be ready to present some of their approximations at the end, and that all are aware of how long they have left.
We assume that each group will record their reasoning, assumptions and calculations in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:
- Every group is given a couple of minutes to report back to the whole class, perhaps focussing on explaining two of their approximations. Those listening can seek clarification and ask questions. After each presentation, those listening are invited to offer positive feedback. Finally, those presenting can suggest how the group could have improved their work on the task.
- Everyone makes a poster to put on display at the front of the room, but only a couple of groups are selected to report back to the whole class. Feedback and suggestions can be given in the same way as above. Additionally, students from the groups which don't present can be invited to share at the end anything they did differently.
- Two people from each group move to join an adjacent group. The two "hosts" explain their some of their calculations and reasoning to the two "visitors". The "visitors" act as critical friends, requiring clear mathematical explanations and justifications. The "visitors" then comment on anything they did differently in their own group.
Key questions
If your focus is effective group work, this list of skills may be helpful (Word, PDF). Ask learners to identify which skills they demonstrated, and which skills they need to develop further.
If your focus is mathematical, these prompts might be useful:
What assumptions have you made?
What other information do you need?
Are there any questions which give an exact answer?
Can you say anything about the accuracy of those answers which aren't exact?
Possible extension
Although this is an approximations question, it is possible to give rigorous bounds on the quantities. Students could be encouraged to give a known upper and lower bound for each quantity, with the focus on providing as tight a bound as possible
Students could also compare their approximations with those of others. Is there a sense in which one of best?
Possible support
By working in groups with clearly assigned roles we are encouraging students to take responsibility for ensuring that everyone understands before the group moves on.